Epsilon=One
08-21-2005, 05:42 PM
Klaus Brauer: On Solitons
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This post is in memoriam:
John Scott Russell (www.CQthus.com/JSRussell) (Russel) [1808-1888]
The soliton, which is an irreducible physical manifestation, is the crux of Pulsoid Theory (www.101123.com/PTis) (PT).
It is the ubiquitous soliton, “s,” that is heuristically symbolized by the Natural function (www.101123.com/NF), Psi, Ψ, x² – x (NF) that accounts for the unique, ellipsoidal geometry/Unimetry (www.101123.com/Uni) of the Pulsoid (www.101123.com/P). It is such that underlies the Natural origin of the integers (www.101123.com/NI).
It is the soliton that connects the sinusoidal equations of Light and the ellipsoidal equations of gravity.
The soliton is a physical manifestation in which “likes” attract; thus, from a single manifestation evolves all the variety of existence. It was this problem of likes attracting (coalescence) that Philip Morrison (www.CQthus.com/PM-NYT-Obit) helped resolve in 1955 by quickly referring to the 1871 publication of C. A . Bjerknes (www.CQthus.com/Bjerknes) concerning hydrodynamical forces at a distance, as described and demonstrated, in detail, by his son, Vilhelm, which is referred to in the 1952 writings of Ludwig Prandtl (www.CQthus.com/Prandtl).
See: Heriot-Watt University, Department of Mathematics (www.CQthus.com/SolitonsHP) for the history of the re-discovery of the soliton in 1973.
Authors note:
All the following information about
solitons, on this page, is courtesy of:
www.usf.uni-osnabrueck.de/~kbrauer/solitons.html,
and, Klaus Brauer’s Home Page. (www.usf.uni-osnabrueck.de/~kbrauer/index.en.html)
Note: The above links are often updated and are more current than the below quotes therefrom. For the serious reader the above links are recommended.
________________________________________
Welcome to Klaus Brauer's SOLITON Page
________________________________________
One of the most exciting phenomena in dealing with non-linear Partial Differential Equations are the Solitons, i.e. solitary waves.
The first person reporting these phenomena was the Scottish engineer John Scott Russell, who described the propagation of a wave in shallow water.
Nowadays we have better knowledge of the underlying mathematical properties. Solitons are the solutions of the famous non-linear Korteweg - de Vries Equation.
A solution to this PDE may be found in using the method of Bäcklund transform.
Korteweg - de Vries Equations
(If no image appears below, "Click" your browser "Refresh" icon.)
http://i.g2d.us/kdv.gif
The solution may be visualized as a 3D Plot and as a Contour Plot (both generated with Mathematica 4.0).
Finally it can be nicely observed by looking at the animated graph, produced as well with Mathematica 4.0.
Analytical solution and graphical representation
of the One Soliton solution.
(If no image appears below, "Click" your browser "Refresh" icon.)
http://i.g2d.us/gr_ani1.gif
It is possible to construct solutions to the Korteweg - de Vries equation which are non-linear superpositions of regular and irregular single solutions.
The interested reader is referred to the book:
Vvedensky, Dimitri D.,"Partial Differential Equations with Mathematica - Chapter 9," Addison-Wesley Publishing Company, Reading, MA, ISBN 0-201-54409-1, 1993
The author of this Web page has written an article (16 pages as a PDF file). The contents points out to some history, presents Vvedensky's solutions, and shows some Mathematica code.
Watch the paper here with Acrobat Reader (www.CQthus.com/Brauer/KdV.pdf), Size: 1031 KB
Download a ZIP version (www.CQthus.com/Zip/KdV.zip) of the PDF file here, Size: 329 KB
This construction method has been performed for two and for three superpositioned solutions. Each of them have a parameter, say b1 and b2 for two waves and b1, b2 and b3 for three waves. The effect is that a wave travels the faster the greater that parameter is - thus overtaking a slower wave.
The two or the three waves preserve their shapes even after the overtaking process.
Analytical solution and graphical representation of the Two Solitons solution
Analytical solution and graphical representation of the Three Solitons solution
Further Information:
A lot of information including the revival of John Scott Russell's experiments are given by Heriot-Watt University (Edinburgh/Scotland). See: Heriot-Watt University, Department of Mathematics (www.CQthus.com/SolitonsHP)
The University of Kyoto/Japan has prepared a Soliton-Lab Art Gallery.
Further Information, especially on the Sine-Gordon-Equation, solved by using the Computer Algebra System Maple is coming from Tver State University in Russia (by the way: Tver is a partner city of Osnabrück), look at the page of Andrey E. Miroshnichenko.
Lots of Links may be found from R. Victor Jones' Soliton Page.
Concerning waves in general, please visit Waves, Waves, Waves.
An easy introduction comes from the Technical University of Denmark
Even a German TV channel has brought solitons to the public.
Real solitons observed in the Strait of Gibraltar.
________________________________________
Update: August 17th, 2004
Zurück zu Klaus Brauers Heimatseite
Back to Klaus Brauer's Homepage (http://www.usf.uni-osnabrueck.de/~kbrauer/index.en.html)
________________________________________
(If no image appears below, "Click" your browser "Refresh" icon.)
http://i.g2d.us/breath2.gif
Please ask questions. :)With questions it’s possible to know if
comments are logical and convincing;
or whether clarification is required......http://1.g2d.us/e.gifhttp://7.g2d.us/e.jpghttp://2.g2d.us/e.gifhttp://3.g2d.us/e.gifhttp://4.g2d.us/e.gifhttp://5.g2d.us/e.gif
..........http://6.g2d.us/e.gif
..........If images don’t display, "click" the Refresh Icon.
(If no image appears below, "Click" your browser "Refresh" icon.)
http://i.g2d.us/jsr0.jpg
This post is in memoriam:
John Scott Russell (www.CQthus.com/JSRussell) (Russel) [1808-1888]
The soliton, which is an irreducible physical manifestation, is the crux of Pulsoid Theory (www.101123.com/PTis) (PT).
It is the ubiquitous soliton, “s,” that is heuristically symbolized by the Natural function (www.101123.com/NF), Psi, Ψ, x² – x (NF) that accounts for the unique, ellipsoidal geometry/Unimetry (www.101123.com/Uni) of the Pulsoid (www.101123.com/P). It is such that underlies the Natural origin of the integers (www.101123.com/NI).
It is the soliton that connects the sinusoidal equations of Light and the ellipsoidal equations of gravity.
The soliton is a physical manifestation in which “likes” attract; thus, from a single manifestation evolves all the variety of existence. It was this problem of likes attracting (coalescence) that Philip Morrison (www.CQthus.com/PM-NYT-Obit) helped resolve in 1955 by quickly referring to the 1871 publication of C. A . Bjerknes (www.CQthus.com/Bjerknes) concerning hydrodynamical forces at a distance, as described and demonstrated, in detail, by his son, Vilhelm, which is referred to in the 1952 writings of Ludwig Prandtl (www.CQthus.com/Prandtl).
See: Heriot-Watt University, Department of Mathematics (www.CQthus.com/SolitonsHP) for the history of the re-discovery of the soliton in 1973.
Authors note:
All the following information about
solitons, on this page, is courtesy of:
www.usf.uni-osnabrueck.de/~kbrauer/solitons.html,
and, Klaus Brauer’s Home Page. (www.usf.uni-osnabrueck.de/~kbrauer/index.en.html)
Note: The above links are often updated and are more current than the below quotes therefrom. For the serious reader the above links are recommended.
________________________________________
Welcome to Klaus Brauer's SOLITON Page
________________________________________
One of the most exciting phenomena in dealing with non-linear Partial Differential Equations are the Solitons, i.e. solitary waves.
The first person reporting these phenomena was the Scottish engineer John Scott Russell, who described the propagation of a wave in shallow water.
Nowadays we have better knowledge of the underlying mathematical properties. Solitons are the solutions of the famous non-linear Korteweg - de Vries Equation.
A solution to this PDE may be found in using the method of Bäcklund transform.
Korteweg - de Vries Equations
(If no image appears below, "Click" your browser "Refresh" icon.)
http://i.g2d.us/kdv.gif
The solution may be visualized as a 3D Plot and as a Contour Plot (both generated with Mathematica 4.0).
Finally it can be nicely observed by looking at the animated graph, produced as well with Mathematica 4.0.
Analytical solution and graphical representation
of the One Soliton solution.
(If no image appears below, "Click" your browser "Refresh" icon.)
http://i.g2d.us/gr_ani1.gif
It is possible to construct solutions to the Korteweg - de Vries equation which are non-linear superpositions of regular and irregular single solutions.
The interested reader is referred to the book:
Vvedensky, Dimitri D.,"Partial Differential Equations with Mathematica - Chapter 9," Addison-Wesley Publishing Company, Reading, MA, ISBN 0-201-54409-1, 1993
The author of this Web page has written an article (16 pages as a PDF file). The contents points out to some history, presents Vvedensky's solutions, and shows some Mathematica code.
Watch the paper here with Acrobat Reader (www.CQthus.com/Brauer/KdV.pdf), Size: 1031 KB
Download a ZIP version (www.CQthus.com/Zip/KdV.zip) of the PDF file here, Size: 329 KB
This construction method has been performed for two and for three superpositioned solutions. Each of them have a parameter, say b1 and b2 for two waves and b1, b2 and b3 for three waves. The effect is that a wave travels the faster the greater that parameter is - thus overtaking a slower wave.
The two or the three waves preserve their shapes even after the overtaking process.
Analytical solution and graphical representation of the Two Solitons solution
Analytical solution and graphical representation of the Three Solitons solution
Further Information:
A lot of information including the revival of John Scott Russell's experiments are given by Heriot-Watt University (Edinburgh/Scotland). See: Heriot-Watt University, Department of Mathematics (www.CQthus.com/SolitonsHP)
The University of Kyoto/Japan has prepared a Soliton-Lab Art Gallery.
Further Information, especially on the Sine-Gordon-Equation, solved by using the Computer Algebra System Maple is coming from Tver State University in Russia (by the way: Tver is a partner city of Osnabrück), look at the page of Andrey E. Miroshnichenko.
Lots of Links may be found from R. Victor Jones' Soliton Page.
Concerning waves in general, please visit Waves, Waves, Waves.
An easy introduction comes from the Technical University of Denmark
Even a German TV channel has brought solitons to the public.
Real solitons observed in the Strait of Gibraltar.
________________________________________
Update: August 17th, 2004
Zurück zu Klaus Brauers Heimatseite
Back to Klaus Brauer's Homepage (http://www.usf.uni-osnabrueck.de/~kbrauer/index.en.html)
________________________________________
(If no image appears below, "Click" your browser "Refresh" icon.)
http://i.g2d.us/breath2.gif
Please ask questions. :)With questions it’s possible to know if
comments are logical and convincing;
or whether clarification is required......http://1.g2d.us/e.gifhttp://7.g2d.us/e.jpghttp://2.g2d.us/e.gifhttp://3.g2d.us/e.gifhttp://4.g2d.us/e.gifhttp://5.g2d.us/e.gif
..........http://6.g2d.us/e.gif
..........If images don’t display, "click" the Refresh Icon.